We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck/ log∗ k pairs of parts which are not -regular, where c, > 0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemerédi’s regularity lemma. In order to gain some control ...

Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n...

Abstract. We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared in various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence as well as provide short proofs of the Sze...

We consider a new description logic ALCIr that extends ALCI with role inclusion axioms of the form R ⊑ QR1 . . . Rm satisfying a certain regularity condition. We prove that concept satisfiability with respect to RBoxes in this logic is ExpTime-hard. We then define a restriction ALCIr− of ALCIr and show that concept satisfiability with respect to RBoxes in ALCIr− is PSpace-complete.

We introduce a permutation analogue of the celebrated Szemerédi Regularity Lemma, and derive a number of consequences. This tool allows us to provide a structural description of permutations which avoid a specified pattern, a result that permutations which scatter small intervals contain all possible patterns of a given size, a proof that every permutation avoiding a specified pattern has a nea...

Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n...

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group Fn 2 . A triangle in Fn 2 is a triple (x, y, z) such that x + y + z = 0. The triangle removal lemma for Fn 2 states that for every ε > 0 there is a δ > 0, such that if a subset A of Fn 2 requires the removal of at least ε · 2n elements to make it triangle-free, then it must contain at l...

In 1975, Szemerédi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemerédi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of l...

Szemerédi’s regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemerédi’s theorem on arithmetic progressions [19], [18]. In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to ...

Szemerédi’s Regularity Lemma [22, 23] is a powerful tool in graph theory. It asserts that all large graphs G admit a bounded partition of E(G), most classes of which are bipartite subgraphs with uniformly distributed edges. The original proof of this result was non-constructive. A constructive proof was given by Alon, Duke, Lefmann, Rödl and Yuster [1], which allows one to efficiently construct...